In differential geometry, the third fundamental form is a surface metric denoted by \( {\displaystyle \mathrm {I\!I\!I} } \). Unlike the second fundamental form, it is independent of the surface normal.
Definition
Let S be the shape operator and M be a smooth surface. Also, let up and vp be elements of the tangent space Tp(M). The third fundamental form is then given by
\( {\displaystyle \mathrm {I\!I\!I} (\mathbf {u} _{p},\mathbf {v} _{p})=S(\mathbf {u} _{p})\cdot S(\mathbf {v} _{p})\,.} \)
Properties
The third fundamental form is expressible entirely in terms of the first fundamental form and second fundamental form. If we let H be the mean curvature of the surface and K be the Gaussian curvature of the surface, we have
\( {\displaystyle \mathrm {I\!I\!I} -2H\mathrm {I\!I} +K\mathrm {I} =0\,.} \)
As the shape operator is self-adjoint, for u,v ∈ Tp(M), we find
\( {\displaystyle \mathrm {I\!I\!I} (u,v)=\langle Su,Sv\rangle =\langle u,S^{2}v\rangle =\langle S^{2}u,v\rangle \,.} \)
See also
Metric tensor
First fundamental form
Second fundamental form
Tautological one-form
Various notions of curvature defined in differential geometry
Differential geometry
of curves
Curvature Torsion of a curve Frenet–Serret formulas Radius of curvature (applications) Affine curvature Total curvature Total absolute curvature
Differential geometry
of surfaces
Principal curvatures Gaussian curvature Mean curvature Darboux frame Gauss–Codazzi equations First fundamental form Second fundamental form Third fundamental form
Riemannian geometry
Curvature of Riemannian manifolds Riemann curvature tensor Ricci curvature Scalar curvature Sectional curvature
Curvature of connections
Curvature form Torsion tensor Cocurvature Holonomy
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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